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Emil Jeřábek
Emil Jeřábek
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Let G$\DeclareMathOperator\rk{rk}$ Let $G$ be a finite metabelian p$p$-group, i.e. the commutator subgroup G'$G'$ of G$G$ is abelian. Then I ask myself under which conditions does the following hold:

          rk(G' \cap Z(G)) \le rk(G/G')   (*)

$$\tag{$*$} \rk(G' \cap Z(G)) \le \rk(G/G')$$

where Z(G)$Z(G)$ is the centre of G$G$. I have constructed examples where (*)$(*)$ does not hold, but in most cases it does hold. Do you know of any results in the literature  ? How high would you guess the percentage of p$p$-groups satisfying (**)$(**)$ in a numerical analysis  ?

Thanks a lot.

Let G be a finite metabelian p-group, i.e. the commutator subgroup G' of G is abelian. Then I ask myself under which conditions does the following hold:

          rk(G' \cap Z(G)) \le rk(G/G')   (*)

where Z(G) is the centre of G. I have constructed examples where (*) does not hold, but in most cases it does hold. Do you know of any results in the literature  ? How high would you guess the percentage of p-groups satisfying (**) in a numerical analysis  ?

Thanks a lot.

$\DeclareMathOperator\rk{rk}$ Let $G$ be a finite metabelian $p$-group, i.e. the commutator subgroup $G'$ of $G$ is abelian. Then I ask myself under which conditions does the following hold:

$$\tag{$*$} \rk(G' \cap Z(G)) \le \rk(G/G')$$

where $Z(G)$ is the centre of $G$. I have constructed examples where $(*)$ does not hold, but in most cases it does hold. Do you know of any results in the literature? How high would you guess the percentage of $p$-groups satisfying $(**)$ in a numerical analysis?

Thanks a lot.

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JSE
JSE
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Tobias Bembom
Tobias Bembom
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Center of finite metabelian p-groups

Let G be a finite metabelian p-group, i.e. the commutator subgroup G' of G is abelian. Then I ask myself under which conditions does the following hold:

          rk(G' \cap Z(G)) \le rk(G/G')   (*)

where Z(G) is the centre of G. I have constructed examples where (*) does not hold, but in most cases it does hold. Do you know of any results in the literature ? How high would you guess the percentage of p-groups satisfying (**) in a numerical analysis ?

Thanks a lot.